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Properties

Label 2-1176-21.5-c1-0-2
Degree $2$
Conductor $1176$
Sign $-0.311 + 0.950i$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 + 1.60i)3-s + (−1.28 + 2.23i)5-s + (−2.17 − 2.06i)9-s + (−1.43 + 0.826i)11-s + 5.71i·13-s + (−2.76 − 3.50i)15-s + (−3.79 − 6.56i)17-s + (−2.58 − 1.49i)19-s + (−0.249 − 0.143i)23-s + (−0.825 − 1.43i)25-s + (4.72 − 2.16i)27-s + 2.05i·29-s + (5.21 − 3.00i)31-s + (−0.409 − 2.83i)33-s + (−0.877 + 1.51i)37-s + ⋯
L(s)  = 1  + (−0.371 + 0.928i)3-s + (−0.576 + 0.998i)5-s + (−0.724 − 0.689i)9-s + (−0.431 + 0.249i)11-s + 1.58i·13-s + (−0.713 − 0.906i)15-s + (−0.919 − 1.59i)17-s + (−0.594 − 0.343i)19-s + (−0.0519 − 0.0300i)23-s + (−0.165 − 0.286i)25-s + (0.908 − 0.417i)27-s + 0.382i·29-s + (0.936 − 0.540i)31-s + (−0.0712 − 0.493i)33-s + (−0.144 + 0.249i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.311 + 0.950i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.311 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1357514821\)
\(L(\frac12)\) \(\approx\) \(0.1357514821\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.642 - 1.60i)T \)
7 \( 1 \)
good5 \( 1 + (1.28 - 2.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.43 - 0.826i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.71iT - 13T^{2} \)
17 \( 1 + (3.79 + 6.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.58 + 1.49i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.249 + 0.143i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.05iT - 29T^{2} \)
31 \( 1 + (-5.21 + 3.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.877 - 1.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.28T + 41T^{2} \)
43 \( 1 - 2.46T + 43T^{2} \)
47 \( 1 + (-0.186 + 0.323i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.73 - 3.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.89 + 8.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.889 + 0.513i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.18 + 2.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + (-3.30 + 1.90i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.56 + 7.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.65T + 83T^{2} \)
89 \( 1 + (7.25 - 12.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.43iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.48056544928321036598026599546, −9.465645699877222781651750486246, −9.031265205327956209628185684665, −7.82230629705767787273246359145, −6.80779935906508107350623456673, −6.41952539036920101994651445167, −4.92950597673553216114658027880, −4.43102874630545057234066651606, −3.34423775673991057921773341778, −2.38072748555930363516942912982, 0.06374540205172949860296177821, 1.29925755110837620334912486338, 2.64947757808049039823215369529, 3.99358557960462542009820420802, 5.03872438374598167638042224731, 5.83362020338801750808931299869, 6.61948113966171533350074086833, 7.86585054479274509555172577838, 8.212262015179503602518657261049, 8.788625868668170576434758672040

Graph of the $Z$-function along the critical line